Find the Exact Value sin(3/8*pi)

Problem

sin((3*π)/8)

Solution

1. Identify the angle as a half-angle of a known value. Since (3*π)/8=1/2⋅(3*π)/4 we can use the half-angle identity for sine.

2. Apply the formula for the half-angle of sine:

sin(θ/2)=±√(,(1−cos(θ))/2)

3. Substitute θ=(3*π)/4 into the formula. Since (3*π)/8 is in the first quadrant, the result must be positive:

sin((3*π)/8)=√(,(1−cos((3*π)/4))/2)

4. Evaluate the cosine of (3*π)/4

cos((3*π)/4)=−√(,2)/2

5. Substitute the cosine value back into the expression:

sin((3*π)/8)=√(,(1−(−√(,2)/2))/2)

6. Simplify the numerator inside the square root:

sin((3*π)/8)=√(,(1+√(,2)/2)/2)

7. Combine the terms in the numerator:

sin((3*π)/8)=√(,(2+√(,2))/2/2)

8. Divide by 2 to reach the final simplified form:

sin((3*π)/8)=√(,(2+√(,2))/4)

9. Simplify the square root of the denominator:

sin((3*π)/8)=√(,2+√(,2))/2

Final Answer

sin((3*π)/8)=√(,2+√(,2))/2

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